WOLFRAM

Wolfram Language & System Documentation Center

FrobeniusSolve[{a1,…,an},b]

gives a list of all solutions of the Frobenius equation .

FrobeniusSolve[{a1,…,an},b,m]

gives at most m solutions.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Tech Notes
Related Guides
Related Links
History
Cite this Page

FrobeniusSolve

FrobeniusSolve[{a1,…,an},b]

gives a list of all solutions of the Frobenius equation .

FrobeniusSolve[{a1,…,an},b,m]

gives at most m solutions.

Details

  • The Frobenius equation is the Diophantine equation , where the ai are positive integers, b is an integer, and a solution must consist of non-negative integers. For negative b there are no solutions.

Examples

open all close all

Basic Examples  (1)

All solutions of the Frobenius equation :

Wolfram Language code: FrobeniusSolve[{12, 16, 20, 27}, 123]

Check:

Wolfram Language code: %.{12, 16, 20, 27}

Scope  (1)

Show that 43 cannot be represented as a sum of positive integer multiples of 6, 9 and 20:

Wolfram Language code: FrobeniusSolve[{6, 9, 20}, 43]

Find all such representations of 44:

Wolfram Language code: FrobeniusSolve[{6, 9, 20}, 44]

Return just a single representation:

Wolfram Language code: FrobeniusSolve[{6, 9, 20}, 44, 1]

Properties & Relations  (2)

Reduce may also be used to find the solutions to the Frobenius equation:

Wolfram Language code: Reduce[{12Subscript[x, 1] + 16Subscript[x, 2] + 20Subscript[x, 3] + 27Subscript[x, 4] == 123, Subscript[x, 1] ≥ 0, Subscript[x, 2] ≥ 0, Subscript[x, 3] ≥ 0, Subscript[x, 4] ≥ 0}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4]}, Integers]

FrobeniusSolve returns the same solution set:

Wolfram Language code: FrobeniusSolve[{12, 16, 20, 27}, 123]

FrobeniusSolve gives coefficient lists for IntegerPartitions:

Wolfram Language code: FrobeniusSolve[{6, 9, 20}, 24]
Wolfram Language code: IntegerPartitions[24, All, {6, 9, 20}]
Wolfram Research (2007), FrobeniusSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusSolve.html.

Text

Wolfram Research (2007), FrobeniusSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusSolve.html.

CMS

Wolfram Language. 2007. "FrobeniusSolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FrobeniusSolve.html.

APA

Wolfram Language. (2007). FrobeniusSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrobeniusSolve.html

BibTeX

@misc{reference.wolfram_2026_frobeniussolve, author="Wolfram Research", title="{FrobeniusSolve}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/FrobeniusSolve.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_frobeniussolve, organization={Wolfram Research}, title={FrobeniusSolve}, year={2007}, url={https://reference.wolfram.com/language/ref/FrobeniusSolve.html}, note=[Accessed: 12-June-2026]}